College Trigonometry & Analytic Geometry Poudre School .
College Trigonometry & Analytic GeometryPoudre School DistrictPacing OverviewSection TitlePacingNotesSemester 1Analytic Geometry/Conics (Chapters 10.1-10.4 & 11.6)10.110.210.310.411.613.5 daysChapter 10.1-10.4: Analytic GeometryConics2 daysThe Parabola2 daysThe Ellipse2 daysThe Hyperbola2.5 daysChapter 11.6: Systems of Equations and InequalitiesSystems of Nonlinear Equations1.5 daysTrigonometric Functions (Chapter 6)6.16.26.36.46.56.613.5 daysAngles and Their MeasureTrigonometric Functions: Unit Circle ApproachProperties of the Trigonometric FunctionsGraphs of the Sine and Cosine FunctionsGraphs of the Tangent, Cotangent, Cosecant, andSecant FunctionsPhase Shift; Sinusoidal Curve Fitting2 days1.5 days2 days1.5 days1.5 days1.5 daysAnalytic Trigonometry (Chapter 7.1-7.3)7.17.27.32017-20185 daysThe Inverse Sine, Cosine, and Tangent FunctionsThe Inverse Trigonometric Functions (continued)Trigonometric Equations1.5 days1 day1.5 days1 Page
College Trigonometry & Analytic GeometryPoudre School DistrictPacing OverviewSection TitlePacingNotesSemester 2Analytic Trigonometry Continued (Chapter 7.4-7.7)7.1-7.37.47.57.67.78 daysReviewPolynomialsSum and Difference FormulasDouble-angle and Half-angle FormulasProduct-to-Sum and Sum-to-Product Formulas0.5 day2 days1.5 days1 day1 dayoptionalApplications of Trig Functions (Chapter 8)8.18.28.38.48.57 daysRight Triangle Trigonometry; ApplicationsThe Law of SinesThe Law of CosinesArea of a TriangleSimple Harmonic Motion Damped Motion;Combining Waves1 day1 day1 day0.5 day1 dayPolar Coordinates (Chapter 9)9.19.29.39.49.59.69.717 daysPolar CoordinatesPolar Equations and GraphsThe Complex Plane; DeMoivre’s TheoremVectorsThe Dot ProductVectors in SpaceThe Cross Product1 day3.5 days2.5 days2 days1 day1 day1 daySequences and Series (Chapter 12)12.112.212.312.412.58 daysSequencesArithmetic SequencesGeometric Sequences; Geometric SeriesMathematical InductionThe Binomial Theorem1 day1.5 days1.5 days1 day1 dayif time allowsIf time allowsNote: The schedule above is for the course and should be completed prior to seniors’ last day.The sections below are reserve for instruction, time permitting.A Preview of Calculus: The Limit, Derivative, and Integral of a Function(Chapter 14)14.114.214.33 daysFinding Limits Using Tables and GraphsAlgebra Techniques for Finding LimitsOne-Sided Limits; Continuous Functions1 day1 day1 dayif time allowsif time allowsif time allowsAnalytic Geometry (Chapter 10.5-10.7)10.510.610.72017-20186 daysRotation of Axes; General Form of a ConicPolar Equations of ConicsPlane Curves and Parametric Equations2 days2 days2 daysif time allowsif time allowsif time allows2 Page
College Trigonometry & Analytic GeometryPoudre School DistrictAnalytic Geometry/Conics (Chapters 10.1-10.4 & 11.6)Chapter SummarySectionTitleLearning ObjectivesPacingChapter 10: Analytic Geometry10.1Conics1. Know the Names of the Conics2 days10.2The Parabola1. Analyze Parabolas with Vertex at theOrigin2. Analyze Parabolas with Vertex at (h, k)3. Solve Applied Problems InvolvingParabolas2 days10.3The Ellipse1. Analyze Ellipses with Center at theOrigin2. Analyze Ellipses with Center at (h, k)3. Solve Applied Problems InvolvingEllipses2 days10.4The Hyperbola1. Analyze Hyperbolas with Center at theOrigin2. Find the Asymptotes of a Hyperbola3. Analyze Hyperbolas with Center at (h, k)4. Solve Applied Problems InvolvingHyperbolas2.5 daysChapter 11: Systems of Equations and Inequalities11.6Systems of NonlinearEquations1. Solve a System of Nonlinear EquationsUsing Substitution2. Solve a System of Nonlinear EquationsUsing Elimination1.5 daysTotal: 13.5 daysNote: Additional days reserved for review and assessment.Things to KnowEllipseHyperbolaParabolaStandardsHS.G-GPE.A.3 ( )2017-2018Derive the equations of ellipses and hyperbolas given the foci, using thefact that the sum or difference of distances from the foci is constant.3 Page
College Trigonometry & Analytic GeometryPoudre School DistrictTrigonometric Functions (Chapter 6)Chapter SummarySectionTitleLearning ObjectivesPacing6.1Angles and TheirMeasure1. Convert between Decimals and Degrees,Minutes, Seconds Measures for Angles2. Find the Length of an Arc of a Circle3. Convert from Degrees to Radians andfrom Radians to Degrees4. Find the Area of a Sector of a Circle5. Find the Linear Speed of an ObjectTraveling in Circular Motion2 days6.2TrigonometricFunctions: Unit CircleApproach1. Find the Exact Values of theTrigonometric Functions Using a Pointon the Unit Circle2. Find the Exact Values of theTrigonometric Functions of QuadrantalAngles3. Find the Exact Values of the𝜋Trigonometric Functions of 45 44. Find the Exact Values of the𝜋Trigonometric Functions of 6 30 and1.5 days𝜋3 60 5. Find the Exact Values of theTrigonometric Functions for Integer𝜋𝜋Multiples of 6 30 , 4 45 , and𝜋 60 36. Use a Calculator to Approximate theValue of a Trigonometric Function7. Use a Circle of Radius r to Evaluate theTrigonometric Functionscontinued on next page2017-20184 Page
College Trigonometry & Analytic GeometryPoudre School DistrictChapter Summary (continued)SectionTitleLearning ObjectivesPacing6.3Properties of theTrigonometricFunctions1. Determine the Domain and the Range ofthe Trigonometric Functions2. Determine the Period of theTrigonometric Functions3. Determine the Signs of theTrigonometric Functions in a GivenQuadrant4. Find the Values of the TrigonometricFunctions Using Fundamental Identities5. Find the Exact Values of theTrigonometric Functions of an AngleGiven One of the Functions and theQuadrant of the Angle6. Use Even-Odd Properties to Find theExact Values of the TrigonometricFunctions2 days6.4Graphs of the Sine andCosine Functions1. Graph Functions of the Form𝑦 𝐴 sin(𝜔𝑥) Using Transformations2. Graph Functions of the Form𝑦 𝐴 cos(𝜔𝑥) Using Transformations3. Determine the Amplitude and Period ofSinusoidal Functions4. Graph Sinusoidal Functions Using KeyPoints5. Find an Equation for a Sinusoidal Graph1.5 days6.5Graphs of the Tangent,Cotangent, Cosecant,and Secant Functions1. Graph Functions of the Form𝑦 𝐴 tan(𝜔𝑥) 𝐵 and𝑦 𝐴 cot(𝜔𝑥) 𝐵2. Graph Functions of the Form𝑦 𝐴 csc(𝜔𝑥) 𝐵 and𝑦 𝐴 sec(𝜔𝑥) 𝐵1.5 days6.6Phase Shift; SinusoidalCurve Fitting1. Graph Sinusoidal Functions of the Form𝑦 𝐴 sin(𝜔𝑥 𝜙) 𝐵2. Build Sinusoidal Models from Data1.5 daysTotal: 13.5 daysNote: Additional days reserved for review and assessment.2017-20185 Page
College Trigonometry & Analytic GeometryPoudre School DistrictThings to Know1 CounterclockwiseRevolution1 Degree (1 )1 RadianAngle in Standard PositionAngular SpeedArc LengthArea of a SectorLinear SpeedPeriodic FunctionTrigonometric FunctionsTrigonometric FunctionsUsing a Circle of Radius rStandardsHS.F-TF.A.3 ( )Use special triangles to determine geometrically the values of sine,cosine, tangent for π/3, π/4 and π/6, and use the unit circle to expressthe values of sine, cosine, and tangent for x, π x, and 2π - x in terms oftheir values for x, where x is any real number.HS.F-TF.A.4 ( )Use the unit circle to explain symmetry (odd and even) and periodicityof trigonometric functions.2017-20186 Page
College Trigonometry & Analytic GeometryPoudre School DistrictAnalytic Trigonometry (Chapter 7.1-7.3)Chapter SummarySectionTitleLearning ObjectivesPacing7.1The Inverse Sine,Cosine, and TangentFunctions1. Find the Exact Value of an Inverse SineFunction2. Find an Approximate Value of anInverse Sine Function3. Use Properties of Inverse Functions toFind Exact Values of Certain CompositeFunctions4. Find the Inverse Function of aTrigonometric Function5. Solve Equations Involving InverseTrigonometric Functions1.5 days7.2The InverseTrigonometricFunctions (continued)1. Find the Exact Value of ExpressionsInvolving the Inverse Sine, Cosine, andTangent Functions2. Define the Inverse Secant, Cosecant andCotangent Functions3. Use a Calculator to Evaluate sec 1 𝑥,csc 1 𝑥, and cot 1 𝑥4. Write a Trigonometric Expression as anAlgebraic Expression1 day7.3TrigonometricEquations1. Solve Equations Involving a SingleTrigonometric Function2. Solve Trigonometric Equations Using aCalculator3. Solve Trigonometric EquationsQuadratic in Form4. Solve Trigonometric Equations UsingFundamental Identities5. Solve Trigonometric Equations Using aGraphing Utility1.5 daysTotal: 5 daysNote: Additional days reserved for review and assessment.2017-20187 Page
College Trigonometry & Analytic GeometryPoudre School DistrictThings to KnowDefinition of the Six InverseTrigonometric FunctionsStandardsHS.F-TF.B.6 ( )Understand that restricting a trigonometric function to a domain onwhich it is always increasing or always decreasing allows its inverse tobe constructed.HS.F-TF.B.7 ( )Use inverse functions to solve trigonometric equations that arise inmodeling contexts; evaluate the solutions using technology, andinterpret them in terms of the context.2017-20188 Page
College Trigonometry & Analytic GeometryPoudre School DistrictAnalytic Trigonometry Continued (Chapter 7.4-7.7)Chapter SummarySection7.1-7.3TitleLearning ObjectivesReviewPacing0.5 day7.4Polynomials1. Use Algebra to Simplify TrigonometricExpressions2. Establish Identities2 days7.5Sum and DifferenceFormulas1. Use Sum and Difference Formulas toFind Exact Values2. Use Sum and Difference Formulas toEstablish Identities3. Use Sum and Difference FormulasInvolving Inverse TrigonometricFunctions4. Solve Trigonometric Equations Linearin Sine and Cosine1.5 days7.6Double-angle andHalf-angle Formulas1. Use Double-angle Formulas to FindExact Values2. Use Double-angle Formulas to EstablishIdentities3. Use Half-angle Formulas to Find ExactValues1 day7.7Product-to-Sum andSum-to-ProductFormulas (optional)1. Express Products as Sums2. Express Sums as Products1 dayTotal: 8 daysNote: Additional days reserved for review and assessment.2017-20189 Page
College Trigonometry & Analytic GeometryPoudre School DistrictThings to KnowDouble-angle FormulasHalf-angle FormulasSum and DifferenceFormulasSum-to-Product FormulasProduct-to-Sum FormulasStandardsHS.F-TF.A.3 ( )Use special triangles to determine geometrically the values of sine,cosine, tangent for π/3, π/4 and π/6, and use the unit circle to expressthe values of sine, cosine, and tangent for x, π x, and 2π - x in terms oftheir values for x, where x is any real number.HS.F-TF.B.7 ( )Use inverse functions to solve trigonometric equations that arise inmodeling contexts; evaluate the solutions using technology, andinterpret them in terms of the context.HS.F-TF.C.9 ( )Prove the addition and subtraction formulas for sine, cosine, andtangent and use them to solve problems.2017-201810 P a g e
College Trigonometry & Analytic GeometryPoudre School DistrictApplications of Trig Functions (Chapter 8)Chapter SummarySectionTitleLearning ObjectivesPacing8.1Right TriangleTrigonometry;Applications1. Find the Value of TrigonometricFunctions of Acute Angles Using RightTriangles2. Use the Complementary Angle Theorem3. Solve Right Triangles4. Solve Applied Problems1 day8.2The Law of Sines1. Solve SAA or ASA Triangles2. Solve SSA Triangles3. Solve Applied Problems1 day8.3The Law of Cosines1. Solve SAS Triangles2. Solve SSS Triangles3. Solve Applied Problems1 day8.4Area of a Triangle1. Find the Area of SAS Triangles2. Find the Area of SSS Triangles8.5Simple HarmonicMotion; DampedMotion; CombiningWaves3. Build a Model for an Object in SimpleHarmonic Motion4. Analyze Simple Harmonic Motion5. Analyze an Object in Damped Motion6. Graph the Sum of Two Functions0.5 day1 dayTotal: 7 daysNote: Additional days reserved for review and assessment.Things to KnowArea of a TriangleLaw of CosinesLaw of SinesStandardsHS.G-SRT.D.9 ( )Derive the formula A 1/2 ab sin(C) for the area of a triangle bydrawing an auxiliary line from a vertex perpendicular to the oppositeside.HS.G-SRT.D.10 ( )Prove the Laws of Sines and Cosines and use them to solve problems.HS.G-SRT.D.11 ( )Understand and apply the Law of Sines and the Law of Cosines to findunknown measurements in right and non-right triangles (e.g.,surveying problems, resultant forces).2017-201811 P a g e
College Trigonometry & Analytic GeometryPoudre School DistrictPolar Coordinates (Chapter 9)Chapter SummarySectionTitleLearning ObjectivesPacing1 day9.1Polar Coordinates1. Plot Points Using Polar Coordinates2. Convert from Polar Coordinates toRectangular Coordinated3. Convert from Rectangular Coordinatesto Polar Coordinates4. Transform Equations between Polar andRectangular Forms9.2Polar Equations andGraphs1. Identify and Graph Polar Equations byConverting o Rectangular Equations2. Test Polar Equations for Symmetry3. Graph Polar Equations by PlottingPoints3.5 days9.3The Complex Plane;DeMoivre’s Theorem1. Plot Points in the Complex Plane2. Convert a Complex Number betweenRectangular Form and Polar Form3. Find Products and Quotients ofComplex Numbers in Polar Form4. Use DeMoivre’s Theorem5. Find Complex Roots2.5 days9.4Vectors1.2.3.4.9.5The Dot Product2017-2018Graph VectorsFind a Position VectorAdd and Subtract Vectors AlgebraicallyFind a Scalar Multiple and theMagnitude of a Vector5. Find a Unit Vector6. Find a Vector from its Direction andMagnitude7. Model with Vectors1. Find the Dot Product of Two Vectors2. Find the Angle between Two Vectors3. Determine Whether Two Vectors areParallel4. Determine Whether Two Vectors areOrthogonal5. Decompose a Vector into TwoOrthogonal Vectors6. Compute Workcontinued on next page2 days1 day12 P a g e
College Trigonometry & Analytic GeometryPoudre School DistrictChapter Summary (continued)SectionTitleLearning ObjectivesPacing9.6Vectors in Space1. Find the Distance between Two Pointsin Space2. Find Position Vectors in Space3. Perform Operations on Vectors4. Find the Dot Product5. Find the Angle between Two Vectors6. Find the Direction Angles of a Vector1 day9.7The Cross Product1. Find the Cross Product of Two Vectors2. Know Algebraic Properties of the CrossProduct3. Know Geometric Properties of the CrossProduct4. Find a Vector Orthogonal to Two GivenVectors5. Find the Area of a Parallelogram1 dayTotal: 17 daysNote: Additional days reserved for review and assessment.2017-201813 P a g e
College Trigonometry & Analytic GeometryPoudre School DistrictThings to KnowAngle 𝜃 Between TwoNonZero Vectors u and vArea of a ParallelogramCross ProductDeMoivre’s TheoremDirection Angles of a Vectorin SpaceDot Productnth Root of a ComplexNumber𝑤 𝑟(cos 𝜃0 𝑖 sin 𝜃0 )Polar Form of a ComplexNumberPosition VectorRelationship Between PolarCoordinates (𝑟, 𝜃) andRectangular Coordinates(𝑥, 𝑦)Unit VectorStandardsHS.N-CN.A.3 ( )Find the conjugate of a complex number; use conjugates to find moduliand quotients of complex numbers.HS.N-CN.B.4 ( )Represent complex numbers on the complex plane in rectangular andpolar form (including real and imaginary numbers), and explain whythe rectangular and polar forms of a given complex number representthe same number.HS.N-CN.B.5 ( )Represent addition, subtraction, multiplication, and conjugation ofcomplex numbers geometrically on the complex plane; use propertiesof this representation for computation. For example, (-1 3 i)3 8because (-1 3 i) has modulus 2 and argument 120 .HS.N-CN.B.6 ( )Calculate the distance between numbers in the complex plane as themodulus of the difference, and the midpoint of a segment as theaverage of the numbers at its endpoints.HS.N-VM.A.1 ( )Recognize vector quantities as having both magnitude and direction.Represent vector quantities by directed line segments, and useappropriate symbols for vectors and their magnitudes (e.g., v, v , v , v).HS.N-VM.A.2 ( )Find the components of a vector by subtracting the coordinates of aninitial point from the coordinates of a terminal point.HS.N-VM.A.3 ( )Solve problems involving velocity and other quantities that can berepresented by vectors.HS.N-VM.B.4 ( )Add and subtract vectors.HS.N-VM.B.4aAdd vectors end-to-end, component-wise, and by the parallelogramrule. Understand that the magnitude of a sum of two vectors is typicallynot the sum of the magnitudes.HS.N-VM.B.4bGiven two vectors in magnitude and direction form, determine themagnitude and direction of their sum.2017-201814 P a g e
College Trigonometry & Analytic GeometryPoudre School DistrictStandards (continued)HS.N-VM.B.4cUnderstand vector subtraction v - w as v (-w), where -w is theadditive inverse of w, with the same magnitude as w and pointing inthe opposite direction. Represent vector subtraction graphically byconnecting the tips in the appropriate order, and perform vectorsubtraction component-wise.HS.N-VM.B.5 ( )Multiply a vector by a scalar.HS.N-VM.B.5aRepresent scalar multiplication graphically by scaling vectors andpossibly reversing their direction; perform scalar multiplicationcomponent-wise, e.g., as c(vx, vy) (cvx, cvy).HS.N-VM.B.5bCompute the magnitude of a scalar multiple cv using cv c v.Compute the direction of cv knowing that when c v 0, the directionof cv is either along v (for c 0) or against v (for c 0).HS.N-VM.C.12 ( )Work with 2 2 matrices as a transformations of the plane, andinterpret the absolute value of the determinant in terms of area.2017-201815 P a g e
College Trigonometry & Analytic GeometryPoudre School DistrictSequences and Series (Chapter 12)Chapter SummarySectionTitleLearning ObjectivesPacing12.1Sequences1. Write the First Several Terms of aSequence2. Write the Terms of a Sequence Definedby a Recursive formula3. Use Summation Notation4. Find the Sum of a Sequence1 day12.2Arithmetic Sequences1. Determine Whether a Sequence isArithmetic2. Find a Formula for an ArithmeticSequence3. Find the Sum of an Arithmetic Sequence1.5 days12.3Geometric Sequences;Geometric Series1. Determine Whether a Sequence isGeometric2. Find a Formula for a GeometricSequence3. Find the Sum of a Geometric Sequence4. Determine Whether a Geometric SeriesConverges or Diverges5. Solve Annuity Problems1.5 days12.4MathematicalInduction (if timeallows)1. Prove Statements Using MathematicalInduction1 day12.5The Binomial Theorem(if time allows)𝑛1. Evaluate ( 𝑗 )2. Use the Binomial Theorem1 dayTotal: 8 daysNote: Additional days reserved for review and assessment.2017-201816 P a g e
College Trigonometry & Analytic GeometryPoudre School DistrictThings to KnowAmount of AnnuityArithmetic SequenceBinomial CoefficientBinomial TheoremFactorialsGeometric SequenceInfinite Geometric SeriesPrinciple of MathematicalInductionSequenceSum of a Convergent InfiniteGeometric SeriesSum of the first n terms of anArithmetic SequenceSum of the first n terms of aGeometric SequenceThe Pascal TriangleStandardsHS.A-APR.C.5 ( )2017-2018Know and apply the Binomial Theorem for the expansion of (x y)n inpowers of x and y for a positive integer n, where x and y are anynumbers, with coefficients determined for example by Pascal'sTriangle.17 P a g e
College Trigonometry & Analytic GeometryPoudre School DistrictA Preview of Calculus: The Limit, Derivative, and Integral of a Function(Chapter 14)(time permitting)Chapter SummarySectionTitleLearning ObjectivesPacing14.1Finding Limits UsingTables and Graphs1. Find the Limit Using a Table2. Find the Limit Using a Graph1 day14.2Algebra Techniques forFinding Limits1. Find the Limit of a Sum, a Difference,and a Product2. Find the Limit of a Polynomial3. Find the Limit of a Power or a Root4. Find the Limit of a Quotient5. Find the Limit of an Average Rate ofChange1 day14.3One-Sided Limits;Continuous Functions1. Find the One-sided Limits of a Function2. Determine Whether a Function isContinuous1 dayTotal: 3 daysNote: Additional days reserved for review and assessment.Things to KnowContinuous FunctionLimitLimit FormulasLimit of a PolynomialLimit PropertiesStandards2017-201818 P a g e
College Trigonometry & Analytic GeometryPoudre School DistrictAnalytic Geometry (Chapter 10.5-10.7)(time permitting)Chapter SummarySectionTitleLearning ObjectivesPacing10.5Rotation of Axes;General Form of aCo
Analytic Trigonometry (Chapter 7.1-7.3) Chapter Summary Section Title Learning Objectives Pacing 7.1 The Inverse Sine, Cosine, and Tangent Functions 1. Find the Exact Value of an Inverse Sine Function 2. Find an Approximate Value of an Invers
Oct 02, 2015 · Origin of Analytic Geometry Return to Table of Contents Slide 5 / 202 Analytic Geometry is a powerful combination of geometry and algebra. Many jobs that are looking for employees now, and will be in the future, rely on the process or results of analytic geometry. This includes jobs in medicine, veterinary science,
(Answers for Chapter 5: Analytic Trigonometry) A.5.1 CHAPTER 5: Analytic Trigonometry SECTION 5.1: FUNDAMENTAL TRIGONOMETRIC IDENTITIES 1) Left Side Right Side Type of Identity (ID) csc(x) 1 sin(x) Reciprocal ID tan(x) 1
COMPLEX ANALYTIC GEOMETRY AND ANALYTIC-GEOMETRIC CATEGORIES YA’ACOV PETERZIL AND SERGEI STARCHENKO Abstract. The notion of a analytic-geometric category was introduced by v.d. Dries and Miller in [4]. It is a category of subsets of real analytic manifolds which extends the c
Cache la Poudre River Natural Areas Management Plan Update 2011 Ch. 1 Purpose, Scope and Process 1 CHAPTER 1 A. Vision for the Poudre River Natural Areas
The Cache la Poudre River: From Snow to Flow The Cache la Poudre River is located in northern Colorado. It spans nearly 130 miles in length and drops over 6,100 feet (1860 meters) in elevation. The river is a popular destination for rafting and fishi
Analytic Geometry Geometry is all about shapes and their properties. If you like playing with objects, or like drawing, then geometry is for you! Geometry can be divided into: Plane Geometry is about flat shapes like lines, circles and triangles . shapes that can be drawn on a piece of paper S
Analytic geometry connects algebra and geometry, resulting in powerful methods of analysis and problem solving. The first unit of Analytic Geometry involves similarity, congruence, and proofs. Students will understand similarity in terms of similarity transformations, prove
2003–2008 Mountain Goat Software Scrum roles and responsibilities Defines the features of the product, decides on release date and content Is responsible for the profitability of the product (ROI) Prioritizes features according to market value Can change features and priority every sprint Accepts or rejects work results Product Owner .
